The s Equation

The two-equation model of Launder and co-workers (22,23) and the one most often used employs the S equation given by... 

In order to complete the closure, the various length scales in the models above must be prescribed or related to the other independent variables through a differential equation. Daly and Harlow use a dynamical equation for S>, derived exactly from the Navier-Stokes equations and then closed by assumptions. The S) equation will be discussed presently. Daly and Harlow are now considering the use of two length-scale equations for the dissipating and energy-containing eddies. 

The equilibria of (4.3) are obtained from those of (3.4) by deleting the S equation and using (4.2) to replace S. In order to conserve notation, we retain the labels Eq,Ei,E2 for the equilibria of (4.3). For convenience, and since we require some of the relations below, we restate the equilibrium conditions here. The steady state Eq is given by... 

Inclusion of the T coefficients results in the creation of several additional groups of diagrams, depending on whether a linear or full CCSD method is considered. The former case provides 9/36 additional diagrammatic terms, compared to LCCD, while the full CCSD produces 69/252 diagrams, i.e., 8 out of 15 classes. These are placed in Table III under S and D entries. It may be seen there what type of mutual interrelation between S and D leads to the fifth-order diagrams. In order to reproduce the fifth-order terms according to the CC scheme, we need to carry out two iterations of the S equation and three iterations of the D equation. 

This is a solubility equation in terms of X alone. The ten pieces of solubility data above should be plenty to determine the five unknown constants in this equation. An effort to do this algebraically is instructive. Kjg and are so small that the precision of the data under the conditions makes it impossible to determine them well. Other data and extrapolation suggest that is about 10" and that Kj, is about 10 M. These mean that [Ag" ] and [AgSCN] are small for the data given, that only the last three terms in the S equation are important ... 

A stable starting point for the kinds of flows enconntered in a stirred tank is the k-8 model. This model assumes that the normal stresses are roughly equal and are adequately represented by k. Two differential eqnations are used to model the production, distribution, and dissipation of tmbulent kinetic energy the k-equation, and the s-equation. These equations were developed for free shear flows, and experimentally determined constants are established for the model parameters. One of these constants is nsed to relate local values of k and e to an estimate of (uv) using a modified tmbulent viscosity approach ... 

In a binary liquid solution containing one noncondensable and one condensable component, it is customary to refer to the first as the solute and to the second as the solvent. Equation (13) is used for the normalization of the solvent s activity coefficient but Equation (14) is used for the solute. Since the normalizations for the two components are not the same, they are said to follow the unsymmetric convention. The standard-state fugacity of the solvent is the fugacity of the pure liquid. The standard-state fugacity of the solute is Henry s constant. 

Figure 4 shows experimental and predicted phase equilibria for the acetonitrile/benzene system at 45°C. This system exhibits moderate positive deviations from Raoult s law. The high-quality data of Brown and Smith (1955) are very well represented by the UNIQUAC equation. 

Figure 5 shows the isothermal data of Edwards (1962) for n-hexane and nitroethane. This system also exhibits positive deviations from Raoult s law however, these deviations are much larger than those shown in Figure 4. At 45°C the mixture shown in Figure 5 is only 15° above its critical solution temperature. Again, representation with the UNIQUAC equation is excellent.

Figure 7 shows a fit of the UNIQUAC equation to the iso-baric data of Nakanishi et al. (1967) for the methanol-diethyl-amine system this system also exhibits strong negative deviations from Raoult s law. The UNIQUAC equation correctly re-... 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

Wittrig, T. S., "The Prediction of Liquid-Liquid Equilibria by the UNIQUAC Equation," B.S. Degree Thesis, University of Illinois, Urbana (1977). 

Dieterici s equation A modification of van der Waals equation, in which account is taken of the pressure gradient at the boundary of the gas. It is written... 

The acentric factor is calculated using Edmister s equation (1948) ... 

Fuller s equation, applied for the estimation of the coefficient of diffusion of a binary gas mixture, at a pressure greater than 10 bar, predicts values that are too high. As a first approximation, the value of the coefficient of diffusion can be corrected by multiplying it by the compressibility of the gas /... 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

It is easy to calculate tire value of e as an application of the MAXWELL S equations in the case of a symmetric tube without defect. 

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

Still another situation is that of a supersaturated or supercooled solution, and straightforward modifications can be made in the preceding equations. Thus in Eq. IX-2, x now denotes the ratio of the actual solute activity to that of the saturated solution. In the case of a nonelectrolyte, x - S/Sq, where S denotes the concentration. Equation IX-13 now contains AH, the molar heat of solution. 

Equation X-17 was stated in qualitative form by Young in 1805 [30], and we will follow its designation as Young s equation. The equivalent equation, Eq. X-19, was stated in algebraic form by Dupre in 1869 [31], along with the definition of work of adhesion. An alternative designation for both equations, which are really the same, is that of the Young and Dupre equation (see Ref. 32 for an emphatic dissent). 

The preceding definitions have been directed toward the treatment of the solid-liquid-gas contact angle. It is also quite possible to have a solid-liquid-liquid contact angle where two mutually immiscible liquids are involved. The same relationships apply, only now more care must be taken to specify the extent of mutual saturations. Thus for a solid and liquids A and B, Young s equation becomes... 

Since both sides of Eq. X-39 can be determined experimentally, from heat of immersion measurements on the one hand and contact angle data, on the other hand, a test of the thermodynamic status of Young s equation is possible. A comparison of calorimetric data for n-alkanes [18] with contact angle data [95] is shown in Fig. X-11. The agreement is certainly encouraging. 

Thus, adding surfactants to minimize the oil-water and solid-water interfacial tensions causes removal to become spontaneous. On the other hand, a mere decrease in the surface tension of the water-air interface, as evidenced, say, by foam formation, is not a direct indication that the surfactant will function well as a detergent. The decrease in yow or ysw implies, through the Gibb s equation (see Section III-5) adsorption of detergent. 

Because of their prevalence in physical adsorption studies on high-energy, powdered solids, type II isotherms are of considerable practical importance. Bmnauer, Emmett, and Teller (BET) [39] showed how to extent Langmuir s approach to multilayer adsorption, and their equation has come to be known as the BET equation. The derivation that follows is the traditional one, based on a detailed balancing of forward and reverse rates. 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

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